Function machines

One common approach to functions is to express them as input-output machines, since any function will only generate one value for any input. For one-to-one functions the machines can be reversed so that outputs can be tracked back to their inputs, and this process can be used to solve linear equations with one unknown on one side by replacing each unary operation with its inverse. The machine approach can also be used to construct inverse function expressions, again by constructing a sequence of inverses.

Critics would say that this approach focuses too much on data which, if plotted as a graph, might encourage a ‘join the dots’ approach to the function concept. Another problem is that arranging the function as a flow diagram can lead to some students believing that all algebra can be read and enacted from left to right.

However, the machine model can also encourage a focus on the relations between inputs and outputs, and this model can aid the construction of algorithms to generate function values. Also, consideration of how changes in input affect changes in output – a covariation approach – can support early calculus and some applications of graphs in other subject areas.